Question

Find the value of k for which the quadratic equation kx^2 + 2x + 1 = 0 has a real and distinct root.

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{quadratic}\mathrm{equation}\mathrm{is}, \\ {\mathrm{kx}}^2+2\mathrm{x}+1=0 \\ \mathrm{Now},\mathrm{a}=\mathrm{k};\mathrm{b}=2;\mathrm{c}=1 \\ \mathrm{Now},\mathrm{D}={\mathrm{b}}^2-4\mathrm{ac} \\ ={\left(2\right)}^2-4\left(\mathrm{k}\right)\left(1\right) \\ =4-4\mathrm{k} \\ \mathrm{For}\mathrm{real}\mathrm{and}\mathrm{distinct}\mathrm{roots}, \\ \mathrm{D}>0 \\ \Rightarrow 4-4\mathrm{k}>0 \\ \Rightarrow 4>4\mathrm{k} \\ \Rightarrow 4\mathrm{k}<\hspace{0.17em}4 \\ \Rightarrow \mathrm{k}<1 \end{gathered}$$